This system arises as a model for various physical phenomena, in particular in the study of standing waves for a mixture of Bose–Einstein condensates of \(\ell \)  different hyperfine states which overlap in space, see for example [1]}. Here we consider the case in which the interaction between particles in the same state is attractive (\(\mu _i>0\) ) and the interaction between particles in any two different states is repulsive (\(\lambda _{ij}<0\) ).
